MBE

In this paper, we consider the evolutionary competition between budding and lytic viral release strategies, using a delay differential equation model with distributed delay. When antibody is not established, the dynamics of competition depends on the respective basic reproductive ratios of the two viruses. If the basic reproductive ratio of budding virus is greater than that of lytic virus and one, budding virus can survive. When antibody is established for both strains but the neutralization capacities are the same for both strains, consequence of the competition also depends only on the basic reproductive ratios of the budding and lytic viruses. Using two concrete forms of the viral production functions, we are also able to conclude that budding virus will outcompete if the rates of viral production, death rates of infected cells and neutralizing capacities of the antibodies are the same for budding and lytic viruses. In this case, budding strategy would have an evolutionary advantage. However, if the antibody neutralization capacity for the budding virus is larger than that for the lytic virus, the lytic virus can outcompete the budding virus provided that its reproductive ratio is very high. An explicit threshold is derived.

DCDS-B

We consider a partial differential equation model that describes the sterile insect release method (SIRM) in a bounded 1-dimensional domain (interval). Unlike everywhere-releasing in the domain as considered in previous works [17] and [14] , we propose the mechanism of releasing on the boundary only. We show existence of the fertile-free steady state and prove its stability under some conditions. By using the upper-lower solution method, we also show that under some other conditions there may exist a coexistence steady state. Biological implications of our mathematical results are that the SIRM with releasing only on the boundary can successfully eradicate the fertile insects as long as the strength of the sterile releasing is reasonably large, while the method may also fail if the releasing is not sufficient.

MBE

In this paper, we modify the classic Ross-Macdonald
model for malaria disease dynamics by incorporating latencies both for human beings and female mosquitoes. One novelty of our model is that we introduce two general probability functions ($P_1(t)$ and $P_2(t)$) to reflect the fact that the latencies differ from individuals to individuals. We justify the well-posedness of the new model, identify the basic reproduction number $\mathcal{R}_0$ for the model and analyze the dynamics of the model. We show that when $\mathcal{R}_0 <1$, the
disease free equilibrium $E_0$ is globally asymptotically stable, meaning that the malaria disease will eventually die out; and if $\mathcal{R}_0 >1$, $E_0$ becomes unstable.
When $\mathcal{R}_0 >1$, we consider two specific forms for $P_1(t)$ and $P_2(t)$: (i) $P_1(t)$ and $P_2(t)$ are both exponential functions; (ii) $P_1(t)$ and $P_2(t)$ are both step functions.
For (i), the model reduces to an ODE system, and for (ii), the long term disease dynamics are governed by a DDE system. In both cases, we are able to show that when $\mathcal{R}_0>1$ then the disease will persist; moreover if there is no recovery ($\gamma_1=0$), then all admissible positive solutions will converge to the unique endemic equilibrium. A significant impact of the latencies is that they reduce the basic reproduction number, regardless of the forms of the distributions.

MBE

We consider a mathematical model that describes the interactions of
the HIV virus, CD4 cells and CTLs within host, which is a
modification of some existing models by incorporating (i) two
distributed kernels reflecting the variance of time for virus to
invade into cells and the variance of time for invaded virions to
reproduce within cells; (ii) a nonlinear incidence function $f$ for
virus infections, and (iii) a nonlinear removal rate function $h$
for infected cells. By constructing Lyapunov functionals and subtle
estimates of the derivatives of these Lyapunov functionals, we shown
that the model has the threshold dynamics: if the basic
reproduction number (BRN) is less than or equal to one, then the
infection free equilibrium is globally asymptotically stable,
meaning that HIV virus will be cleared; whereas if the BRN is larger
than one, then there exist an infected equilibrium which is globally
asymptotically stable, implying that the HIV-1 infection will
persist in the host and the viral concentration will approach a
positive constant level. This together with the
dependence/independence of the BRN on $f$ and $h$ reveals the effect
of the adoption of these nonlinear functions.

MBE

We consider a neuronal network model with both axonal connections (in the form of
synaptic coupling) and delayed non-local feedback connections. The kernel in the
feedback channel is assumed to be a standard non-local one, while for
the kernel in the synaptic coupling, four types are considered.
The main concern is the existence of travelling wave front. By employing
the speed index function, we are able to obtain the existence of a travelling wave
front for each of these four types within certain range of model parameters.
We are also able to describe how the feedback coupling strength and
the magnitude of the delay affect the wave speed. Some particular kernel functions
for these four cases are chosen to numerically demonstrate the
theoretical results.

DCDS-B

A sufficient condition is established for globally asymptotic stability
of the positive equilibrium of a regulated logistic growth model with a
delay in the state feedback. The result improves some existing criteria for this
model. It is in a form that is related to the number $3/2$ and the coupling
strength, and thus, is comparable to the well-known $3/2$ condition for the uncontrolled
delayed logistic equation. The comparison seems to suggest that
the mechanism of the control in this model might be inappropriate and new
mechanism should be introduced.

DCDS-B

In this paper, we investigate the cost of immunological up- regulation caused by infection in a between-host transmission dynamical model with superinfection. After introducing a mutant host to an existing model, we explore this problem in (A) monomorphic case and (B) dimorphic case. For (A), we assume that only strain 1 parasite can infect the mutant host. We identify an appropriate fitness for the invasion of the mutant host by analyzing the local stability of the mutant free equilibrium. After specifying a trade-off between the production and recovery rates of infected hosts, we employ the adaptive dynamical approach to analyze the evolutionary and convergence stabilities of the corresponding singular strategy, leading to some conditions for continuously stable strategy, evolutionary branching point and repeller. For (B), a new fitness is introduced to measure the invasion of mutant host under the assumption that both parasite strains can infect the mutant host. By considering two trade-off functions, we can study the conditions for evolutionary stability, isoclinic stability and absolute convergence stability of the singular strategy. Our results show that the host evolution would not favour high degree of immunological up-regulation; moreover, superinfection would help the parasite with weaker virulence persist in hosts.

DCDS-B

In this paper, we consider a mathematical model for HIV-1
infection with intracellular delay and cell-mediated immune
response. A novel feature is that both cytotoxic T lymphocytes
(CTLs) and the intracellular delay are incorporated into the
model. We obtain a necessary and sufficient condition for the
global stability of the infection-free equilibrium and give
sufficient conditions for the local stability of the two infection
equilibria: one without CTLs being activated and the other with.
We also perform some numerical simulations which support the
obtained theoretical results. These results show that larger
intracellular delay may help eradicate the virus, while the
activation of CTLs can only help reduce the virus load and
increase the healthy CD$_4^+$ cells population in the long term
sense.

DCDS-B

A two-patch predator-prey model with the Holling type II functional response is studied, in which
predators are assumed to adopt adaptive dispersal to inhabit
the better patch in order to gain more fitness. Analytical conditions for
the persistence and extinction of predators are obtained under different scenarios of the model. Numerical simulations are conducted which show
that adaptive dispersal can stabilize the system with either weak or strong adaptation,
when prey and predators tend to a globally stable equilibrium in one isolated patch and tend to limit cycles in the other.
Furthermore, it is observed that the adaptive dispersal may cause torus bifurcation for the model when the prey and predators population tend to limit cycles in each isolated patch.

DCDS

In this paper, we establish the existence of traveling
wavefronts for delayed reaction diffusion systems without quasimonotonicity
in the reaction term, by using Schauder's fixed point theorem. We show the
merit of our result by applying it to
the Belousov-Zhabotinskii reaction model with two delays.